In the presence of multicollinearity in logistic regression, the variance of the Maximum Likelihood Estimator (MLE) becomes inflated. Siray et al. (2015) [1] proposed a restricted Liu estimator in logistic regression model with exact linear restrictions. However, there are some situations, where the linear restrictions are stochastic. In this paper, we propose a Stochastic Restricted Maximum Likelihood Estimator (SRMLE) for the logistic regression model with stochastic linear restrictions to overcome this issue. Moreover, a Monte Carlo simulation is conducted for comparing the performances of the MLE, Restricted Maximum Likelihood Estimator (RMLE), Ridge Type Logistic Estimator(LRE), Liu Type Logistic Estimator(LLE), and SRMLE for the logistic regression model by using Scalar Mean Squared Error (SMSE).
In many fields of study such as medicine and epidemiology, it is very important to predict a binary response variable, or to compute the probability of occurrence of an event, in terms of the values of a set of explanatory variables related to it. For example, the probability of suffering a heart attack is computed in terms of the levels of a set of risk factors such as cholesterol and blood pressure. The logistic regression model serves admirably this purpose and is the most used for these cases.
The general form of logistic regression model is
which follows Bernoulli distribution with parameter
where
where
As many authors have stated (Hosmer and Lemeshow (1989) [
To overcome the problem of multi-collinearity in the logistic regression, many estimators are proposed alternatives to the MLE. The most popular way to deal with this problem is called the Ridge Logistic Regression (RLR), which is first proposed by Schaffer et al. (1984) [
An alternative technique to resolve the multi-collinearity problem is to consider parameter estimation with priori available linear restrictions on the unknown parameters, which may be exact or stochastic. That is, in some practical situations there exist different sets of prior information from different sources like past experience or long association of the experimenter with the experiment and similar kind of experiments conducted in the past. If the exact linear restrictions are available in addition to logistic regression model, many authors propose different estimators for the respective parameter
In this paper we propose a new estimator which is called as the Stochastic Restricted Maximum Likelihood Estimator (SRMLE) when the linear stochastic restrictions are available in addition to the logistic regression model. The rest of the paper is organized as follows. The proposed estimator and its asymptotic properties are given in Section 2. In Section 3, the mean square error matrix and the scalar mean square error for this new estimator are obtained. Section 4 describes some important existing estimators for the logistic regression models. Performance of the proposed estimator with respect to Scalar Mean Squared Error (SMSE) is compared with some existing estimators by performing a Monte Carlo simulation study in Section 5. The conclusion of the study is presented in Section 6.
First consider the multiple linear regression model
where y is an
The Ordinary Least Square Estimator (OLSE) of
where
In addition to sample model (5), consider the following linear stochastic restriction on the parameter space
where r is an
The Restricted Ordinary Least Square Estimator (ROLSE) due to exact prior restriction (i.e.
Theil and Goldberger (1961) [
Suppose that the following linear prior information is given in addition to the general logistic regression model (1)
where h is an
to be known
Duffy and Santner (1989) [
Following RMLE in (11) and the Mixed Estimator (ME) in (9) in the Linear Regression Model, we propose a new estimator which is named as the Stochastic Restricted Maximum Likelihood Estimator (SRMLE) when the linear stochastic restriction (10) is available in addition to the logistic regression model (1).
Asymptotic Properties of SRMLE:
The
The asymtotic covariance matrix of SRMLE equals
To compare different estimators with respect to the same parameter vector
where
The Scalar Mean Square Error (SMSE) of the estimator
For two given estimators
The MSE and SMSE of the proposed estimator SRMLE is
Note that the difference given in (20) is non-negative definite. Thus by the MSE criteria it follows that
To examine the performance of the proposed estimator SRMLE over some existing estimators, the following estimators are considered.
1) Logistic Ridge Estimator
Schaefer et al. (1984) [
where
The asymptotic MSE and SMSE of
where
2) Logistic Liu Estimator
Following Liu (1993) [
where
The asymptotic MSE and SMSE of
where
3) Restricted MLE
As we mentioned in Section 2, Duffy and Santner (1989) [
The asymptotic MSE and SMSE of
where
and
Mean Squared Error Comparisons
・ SRMLE versus LRE
where
Theorem 1 (see Appendix 1), it is clear that
Theorem 4.1. The estimator SRMLE is superior to LRE if and only if
・ SRMLE Versus LLE
where
nite matrices. Further by Theorem 1 (see Appendix 1), it is clear that
Theorem 4.2. The estimator
・ SRMLE versus RMLE
where
Theorem 4.3. The estimator
Based on the above results one can say that the new estimator SRMLE is superior to the other estimators with respect to the mean squared error matrix sense under certain conditions. To check the superiority of the estimators numerically, we then consider a simulation study in the next section.
A Monte Carlo simulation is done to illustrate the performance of the new estimator SRMLE over the MLE, RMLE, LRE, and LLE by means of Scalar Mean Square Error (SMSE). Following McDonald and Galarneau (1975) [
where
noulli (
Moreover, for the restriction, we choose
Further for the ridge parameter k and the Liu parameter d, some selected values are chosen so that
The experiment is replicated 3000 times by generating new pseudo-random numbers and the estimated SMSE is obtained as
The simulation results are listed in Tables A1-A16 (Appendix 3) and also displayed in Figures A1-A4 (Appendix 2). From Figures A1-A4, it can be noticed that in general increase in degree of correlation between two explanatory variables
In this research, we introduced the Stochastic Restricted Maximum Likelihood Estimator (SRMLE) for logistic regression model when the linear stochastic restriction was available. The performances of the SRMLE over MLE, LRE, RMLE, and LLE in logistic regression model were investigated by performing a Monte Carlo simulation study. The research had been done by considering different degree of correlations, different numbers of observations and different values of parameters k, d. It was noted that the SMSE of the MLE was inflated when the multicollinearity was presented and it was severe particularly for small samples. The simulation results showed that the proposed estimator SRMLE had smaller SMSE than the estimator MLE with respect to all the values of n and
We thank the editor and the referee for their comments and suggestions, and the Postgraduate Institute of Science, University of Peradeniya, Sri Lanka for providing necessary facilities to complete this research.
VarathanNagarajah,PushpakanthieWijekoon,11, (2015) Stochastic Restricted Maximum Likelihood Estimator in Logistic Regression Model. Open Journal of Statistics,05,837-851. doi: 10.4236/ojs.2015.57082
Theorem 1. Let A:
Lemma 1. Let the two
k, d = 0.0 | k, d = 0.1 | k, d = 0.2 | k, d = 0.3 | k, d = 0.4 | k, d = 0.5 | k, d = 0.6 | k, d = 0.7 | k, d = 0.8 | k, d = 0.9 | k, d = 0.99 | k, d = 1.0 | |
---|---|---|---|---|---|---|---|---|---|---|---|---|
MLE | 2.6097 | 2.6097 | 2.6097 | 2.6097 | 2.6097 | 2.6097 | 2.6097 | 2.6097 | 2.6097 | 2.6097 | 2.6097 | 2.6097 |
LRE | 2.6097 | 2.1774 | 1.8688 | 1.6361 | 1.4543 | 1.3084 | 1.1892 | 1.0901 | 1.0068 | 0.9361 | 0.8812 | 0.8755 |
RMLE | 2.2682 | 2.2682 | 2.2682 | 2.2682 | 2.2682 | 2.2682 | 2.2682 | 2.2682 | 2.2682 | 2.2682 | 2.2682 | 2.2682 |
LLE | 0.8755 | 0.9995 | 1.1355 | 1.2835 | 1.4435 | 1.6156 | 1.7997 | 1.9958 | 2.2039 | 2.4240 | 2.6325 | 2.6562 |
SRMLE | 1.2274 | 1.2274 | 1.2274 | 1.2274 | 1.2274 | 1.2274 | 1.2274 | 1.2274 | 1.2274 | 1.2274 | 1.2274 | 1.2274 |
k, d = 0.0 | k, d = 0.1 | k, d = 0.2 | k, d = 0.3 | k, d = 0.4 | k, d = 0.5 | k, d = 0.6 | k, d = 0.7 | k, d = 0.8 | k, d = 0.9 | k, d = 0.99 | k, d = 1.0 | |
---|---|---|---|---|---|---|---|---|---|---|---|---|
MLE | 3.7509 | 3.7509 | 3.7509 | 3.7509 | 3.7509 | 3.7509 | 3.7509 | 3.7509 | 3.7509 | 3.7509 | 3.7509 | 3.7509 |
LRE | 3.7509 | 2.8786 | 2.3312 | 1.9525 | 1.6750 | 1.4633 | 1.2971 | 1.1637 | 1.0548 | 0.9646 | 0.8960 | 0.8890 |
RMLE | 2.2452 | 2.2452 | 2.2452 | 2.2452 | 2.2452 | 2.2452 | 2.2452 | 2.2452 | 2.2452 | 2.2452 | 2.2452 | 2.2452 |
LLE | 0.8890 | 1.0689 | 1.2733 | 1.5023 | 1.7558 | 2.0340 | 2.3367 | 2.6640 | 3.0158 | 3.3923 | 3.7521 | 3.7933 |
SRMLE | 1.4179 | 1.4179 | 1.4179 | 1.4179 | 1.4179 | 1.4179 | 1.4179 | 1.4179 | 1.4179 | 1.4179 | 1.4179 | 1.4179 |
k, d = 0.0 | k, d = 0.1 | k, d = 0.2 | k, d = 0.3 | k, d = 0.4 | k, d = 0.5 | k, d = 0.6 | k, d = 0.7 | k, d = 0.8 | k, d = 0.9 | k, d = 0.99 | k, d = 1.0 | |
---|---|---|---|---|---|---|---|---|---|---|---|---|
MLE | 7.2447 | 7.2447 | 7.2447 | 7.2447 | 7.2447 | 7.2447 | 7.2447 | 7.2447 | 7.2447 | 7.2447 | 7.2447 | 7.2447 |
LRE | 7.2447 | 4.4635 | 3.1455 | 2.4005 | 1.9193 | 1.5859 | 1.3436 | 1.1611 | 1.0199 | 0.9083 | 0.8267 | 0.8186 |
RMLE | 2.2263 | 2.2263 | 2.2263 | 2.2263 | 2.2263 | 2.2263 | 2.2263 | 2.2263 | 2.2263 | 2.2263 | 2.2263 | 2.2263 |
LLE | 0.8186 | 1.1287 | 1.5135 | 1.9731 | 2.5075 | 3.1165 | 3.8003 | 4.5589 | 5.3922 | 6.3002 | 7.1813 | 7.2829 |
SRMLE | 1.7693 | 1.7693 | 1.7693 | 1.7693 | 1.7693 | 1.7693 | 1.7693 | 1.7693 | 1.7693 | 1.7693 | 1.7693 | 1.7693 |
k, d=0.0 | k, d=0.1 | k, d=0.2 | k, d=0.3 | k, d=0.4 | k, d=0.5 | k, d=0.6 | k, d=0.7 | k, d=0.8 | k, d=0.9 | k, d=0.99 | k, d=1.0 | |
---|---|---|---|---|---|---|---|---|---|---|---|---|
MLE | 70.5890 | 70.5890 | 70.5890 | 70.5890 | 70.5890 | 70.5890 | 70.5890 | 70.5890 | 70.5890 | 70.5890 | 70.5890 | 70.5890 |
LRE | 70.5890 | 6.5671 | 2.6098 | 1.4620 | 0.9711 | 0.7153 | 0.5650 | 0.4692 | 0.4045 | 0.3589 | 0.3288 | 0.3259 |
RMLE | 2.2118 | 2.2118 | 2.2118 | 2.2118 | 2.2118 | 2.2118 | 2.2118 | 2.2118 | 2.2118 | 2.2118 | 2.2118 | 2.2118 |
LLE | 0.3259 | 1.5179 | 4.0071 | 7.7935 | 12.8770 | 19.2580 | 26.9360 | 35.9120 | 46.1840 | 57.7540 | 69.2758 | 70.6209 |
SRMLE | 2.5410 | 2.5410 | 2.5410 | 2.5410 | 2.5410 | 2.5410 | 2.5410 | 2.5410 | 2.5410 | 2.5410 | 2.5410 | 2.5410 |
k, d = 0.0 | k, d = 0.1 | k, d = 0.2 | k, d = 0.3 | k, d = 0.4 | k, d = 0.5 | k, d = 0.6 | k, d = 0.7 | k, d = 0.8 | k, d = 0.9 | k, d = 0.99 | k, d = 1.0 | |
---|---|---|---|---|---|---|---|---|---|---|---|---|
MLE | 0.8648 | 0.8648 | 0.8648 | 0.8648 | 0.8648 | 0.8648 | 0.8648 | 0.8648 | 0.8648 | 0.8648 | 0.8648 | 0.8648 |
LRE | 0.8648 | 0.8210 | 0.7811 | 0.7447 | 0.7113 | 0.6807 | 0.6525 | 0.6266 | 0.6025 | 0.5803 | 0.5617 | 0.5597 |
RMLE | 2.1057 | 2.1057 | 2.1057 | 2.1057 | 2.1057 | 2.1057 | 2.1057 | 2.1057 | 2.1057 | 2.1057 | 2.1057 | 2.1057 |
LLE | 0.5597 | 0.5875 | 0.6161 | 0.6454 | 0.6756 | 0.7065 | 0.7383 | 0.7708 | 0.8042 | 0.8383 | 0.8697 | 0.8732 |
SRMLE | 0.6141 | 0.6141 | 0.6141 | 0.6141 | 0.6141 | 0.6141 | 0.6141 | 0.6141 | 0.6141 | 0.6141 | 0.6141 | 0.6141 |
k, d = 0.0 | k, d = 0.1 | k, d = 0.2 | k, d = 0.3 | k, d = 0.4 | k, d = 0.5 | k, d = 0.6 | k, d = 0.7 | k, d = 0.8 | k, d = 0.9 | k, d = 0.99 | k, d = 1.0 | |
---|---|---|---|---|---|---|---|---|---|---|---|---|
MLE | 1.2320 | 1.2320 | 1.2320 | 1.2320 | 1.2320 | 1.2320 | 1.2320 | 1.2320 | 1.2320 | 1.2320 | 1.2320 | 1.2320 |
LRE | 1.2320 | 1.1399 | 1.0593 | 0.9882 | 0.9252 | 0.8690 | 0.8186 | 0.7733 | 0.7324 | 0.6953 | 0.6648 | 0.6600 |
RMLE | 2.0967 | 2.0967 | 2.0967 | 2.0967 | 2.0967 | 2.0967 | 2.0967 | 2.0967 | 2.0967 | 2.0967 | 2.0967 | 2.0967 |
LLE | 0.6615 | 0.7101 | 0.7607 | 0.8134 | 0.8682 | 0.9250 | 0.9839 | 1.0448 | 1.1077 | 1.1728 | 1.2330 | 1.2398 |
SRMLE | 0.7675 | 0.7675 | 0.7675 | 0.7675 | 0.7675 | 0.7675 | 0.7675 | 0.7675 | 0.7675 | 0.7675 | 0.7675 | 0.7675 |
k, d = 0.0 | k, d = 0.1 | k, d = 0.2 | k, d = 0.3 | k, d = 0.4 | k, d = 0.5 | k, d = 0.6 | k, d = 0.7 | k, d = 0.8 | k, d = 0.9 | k, d = 0.99 | k, d = 1.0 | |
---|---|---|---|---|---|---|---|---|---|---|---|---|
MLE | 2.3557 | 2.3557 | 2.3557 | 2.3557 | 2.3557 | 2.3557 | 2.3557 | 2.3557 | 2.3557 | 2.3557 | 2.3557 | 2.3557 |
LRE | 2.3557 | 2.0182 | 1.7556 | 1.5460 | 1.3754 | 1.2343 | 1.1161 | 1.0158 | 0.9301 | 0.8560 | 0.7976 | 0.7916 |
RMLE | 2.0892 | 2.0892 | 2.0892 | 2.0892 | 2.0892 | 2.0892 | 2.0892 | 2.0892 | 2.0892 | 2.0892 | 2.0892 | 2.0892 |
LLE | 0.7916 | 0.9067 | 1.0313 | 1.1651 | 1.3082 | 1.4607 | 1.6226 | 1.7937 | 1.9742 | 2.1640 | 2.3428 | 2.3631 |
SRMLE | 1.0961 | 1.0961 | 1.0961 | 1.0961 | 1.0961 | 1.0961 | 1.0961 | 1.0961 | 1.0961 | 1.0961 | 1.0961 | 1.0961 |
k, d = 0.0 | k, d = 0.1 | k, d = 0.2 | k, d = 0.3 | k, d = 0.4 | k, d = 0.5 | k, d = 0.6 | k, d = 0.7 | k, d = 0.8 | k, d = 0.9 | k, d = 0.99 | k, d = 1.0 | |
---|---|---|---|---|---|---|---|---|---|---|---|---|
MLE | 22.7202 | 22.7202 | 22.7202 | 22.7202 | 22.7202 | 22.7202 | 22.7202 | 22.7202 | 22.7202 | 22.7202 | 22.7202 | 22.7202 |
LRE | 22.7202 | 7.1920 | 3.6283 | 2.2157 | 1.5085 | 1.1030 | 0.8486 | 0.6784 | 0.5590 | 0.4719 | 0.4124 | 0.4066 |
RMLE | 2.0835 | 2.0835 | 2.0835 | 2.0835 | 2.0835 | 2.0835 | 2.0835 | 2.0835 | 2.0835 | 2.0835 | 2.0835 | 2.0835 |
LLE | 0.4066 | 1.0428 | 2.0336 | 3.3790 | 5.0791 | 7.1337 | 9.5430 | 12.3070 | 15.4255 | 18.8990 | 22.3278 | 22.7265 |
SRMLE | 2.1982 | 2.1982 | 2.1982 | 2.1982 | 2.1982 | 2.1982 | 2.1982 | 2.1982 | 2.1982 | 2.1982 | 2.1982 | 2.1982 |
k, d = 0.0 | k, d = 0.1 | k, d = 0.2 | k, d = 0.3 | k, d = 0.4 | k, d = 0.5 | k, d = 0.6 | k, d = 0.7 | k, d = 0.8 | k, d = 0.9 | k, d = 0.99 | k, d = 1.0 | |
---|---|---|---|---|---|---|---|---|---|---|---|---|
MLE | 0.5544 | 0.5544 | 0.5544 | 0.5544 | 0.5544 | 0.5544 | 0.5544 | 0.5544 | 0.5544 | 0.5544 | 0.5544 | 0.5544 |
LRE | 0.5544 | 0.5368 | 0.5202 | 0.5046 | 0.4898 | 0.4758 | 0.4626 | 0.4501 | 0.4382 | 0.4271 | 0.4174 | 0.4164 |
RMLE | 2.0701 | 2.0701 | 2.0701 | 2.0701 | 2.0701 | 2.0701 | 2.0701 | 2.0701 | 2.0701 | 2.0701 | 2.0701 | 2.0701 |
LLE | 0.4164 | 0.4295 | 0.4429 | 0.4565 | 0.4703 | 0.4844 | 0.4987 | 0.5135 | 0.5280 | 0.5430 | 0.5567 | 0.5582 |
SRMLE | 0.4368 | 0.4368 | 0.4368 | 0.4368 | 0.4368 | 0.4368 | 0.4368 | 0.4368 | 0.4368 | 0.4368 | 0.4368 | 0.4368 |
k, d = 0.0 | k, d = 0.1 | k, d = 0.2 | k, d = 0.3 | k, d = 0.4 | k, d = 0.5 | k, d = 0.6 | k, d = 0.7 | k, d = 0.8 | k, d = 0.9 | k, d = 0.99 | k, d = 1.0 | |
---|---|---|---|---|---|---|---|---|---|---|---|---|
MLE | 0.7833 | 0.7833 | 0.7833 | 0.7833 | 0.7833 | 0.7833 | 0.7833 | 0.7833 | 0.7833 | 0.7833 | 0.7833 | 0.7833 |
LRE | 0.7833 | 0.7511 | 0.7169 | 0.6853 | 0.6561 | 0.6289 | 0.6037 | 0.5803 | 0.5584 | 0.5380 | 0.5208 | 0.5189 |
RMLE | 2.0642 | 2.0642 | 2.0642 | 2.0642 | 2.0642 | 2.0642 | 2.0642 | 2.0642 | 2.0642 | 2.0642 | 2.0642 | 2.0642 |
LLE | 0.5189 | 0.5433 | 0.5684 | 0.5941 | 0.6204 | 0.6474 | 0.6750 | 0.7032 | 0.7321 | 0.7617 | 0.7888 | 0.7918 |
SRMLE | 0.5620 | 0.5620 | 0.5620 | 0.5620 | 0.5620 | 0.5620 | 0.5620 | 0.5620 | 0.5620 | 0.5620 | 0.5620 | 0.5620 |
k, d = 0.0 | k, d = 0.1 | k, d = 0.2 | k, d = 0.3 | k, d = 0.4 | k, d = 0.5 | k, d = 0.6 | k, d = 0.7 | k, d = 0.8 | k, d = 0.9 | k, d = 0.99 | k, d = 1.0 | |
---|---|---|---|---|---|---|---|---|---|---|---|---|
MLE | 1.5040 | 1.5040 | 1.5040 | 1.5040 | 1.5040 | 1.5040 | 1.5040 | 1.5040 | 1.5040 | 1.5040 | 1.5040 | 1.5040 |
LRE | 1.5040 | 1.3650 | 1.2461 | 1.1434 | 1.0541 | 0.9756 | 0.9064 | 0.8451 | 0.7904 | 0.7414 | 0.7015 | 0.6973 |
RMLE | 2.0593 | 2.0593 | 2.0593 | 2.0593 | 2.0593 | 2.0593 | 2.0593 | 2.0593 | 2.0593 | 2.0593 | 2.0593 | 2.0593 |
LLE | 0.6973 | 0.7632 | 0.8324 | 0.9049 | 0.9809 | 1.0602 | 1.1429 | 1.2290 | 1.3184 | 1.4112 | 1.4976 | 1.5074 |
SRMLE | 0.8479 | 0.8479 | 0.8479 | 0.8479 | 0.8479 | 0.8479 | 0.8479 | 0.8479 | 0.8479 | 0.8479 | 0.8479 | 0.8479 |
k, d = 0.0 | k, d = 0.1 | k, d = 0.2 | k, d = 0.3 | k, d = 0.4 | k, d = 0.5 | k, d = 0.6 | k, d = 0.7 | k, d = 0.8 | k, d = 0.9 | k, d = 0.99 | k, d = 1.0 | |
---|---|---|---|---|---|---|---|---|---|---|---|---|
MLE | 14.4762 | 14.4762 | 14.4762 | 14.4762 | 14.4762 | 14.4762 | 14.4762 | 14.4762 | 14.4762 | 14.4762 | 14.4762 | 14.4762 |
LRE | 14.4762 | 6.4739 | 3.7327 | 2.4479 | 1.7388 | 1.3051 | 1.0201 | 0.8226 | 0.6801 | 0.5738 | 0.4997 | 0.4925 |
RMLE | 2.0555 | 2.0555 | 2.0555 | 2.0555 | 2.0555 | 2.0555 | 2.0555 | 2.0555 | 2.0555 | 2.0555 | 2.0555 | 2.0555 |
LLE | 0.4925 | 0.9960 | 1.6984 | 2.5998 | 3.7000 | 4.9992 | 6.4974 | 8.1944 | 10.0904 | 12.1850 | 14.2409 | 14.4792 |
SRMLE | 2.0212 | 2.0212 | 2.0212 | 2.0212 | 2.0212 | 2.0212 | 2.0212 | 2.0212 | 2.0212 | 2.0212 | 2.0212 | 2.0212 |
k, d = 0.0 | k, d = 0.1 | k, d = 0.2 | k, d = 0.3 | k, d = 0.4 | k, d = 0.5 | k, d = 0.6 | k, d = 0.7 | k, d = 0.8 | k, d = 0.9 | k, d = 0.99 | k, d = 1.0 | |
---|---|---|---|---|---|---|---|---|---|---|---|---|
MLE | 0.4083 | 0.4083 | 0.4083 | 0.4083 | 0.4083 | 0.4083 | 0.4083 | 0.4083 | 0.4083 | 0.4083 | 0.4083 | 0.4083 |
LRE | 0.4083 | 0.3989 | 0.3899 | 0.3812 | 0.3730 | 0.3651 | 0.3574 | 0.3502 | 0.3431 | 0.3364 | 0.3306 | 0.3300 |
RMLE | 2.0524 | 2.0524 | 2.0524 | 2.0524 | 2.0524 | 2.0524 | 2.0524 | 2.0524 | 2.0524 | 2.0524 | 2.0524 | 2.0524 |
LLE | 0.3300 | 0.3376 | 0.3453 | 0.3531 | 0.3611 | 0.3691 | 0.3772 | 0.3853 | 0.3936 | 0.4020 | 0.4096 | 0.4105 |
SRMLE | 0.3400 | 0.3400 | 0.3400 | 0.3400 | 0.3400 | 0.3400 | 0.3400 | 0.3400 | 0.3400 | 0.3400 | 0.3400 | 0.3400 |
k, d = 0.0 | k, d = 0.1 | k, d = 0.2 | k, d = 0.3 | k, d = 0.4 | k, d = 0.5 | k, d = 0.6 | k, d = 0.7 | k, d = 0.8 | k, d = 0.9 | k, d = 0.99 | k, d = 1.0 | |
---|---|---|---|---|---|---|---|---|---|---|---|---|
MLE | 0.5801 | 0.5801 | 0.5801 | 0.5801 | 0.5801 | 0.5801 | 0.5801 | 0.5801 | 0.5801 | 0.5801 | 0.5801 | 0.5801 |
LRE | 0.5801 | 0.5602 | 0.5413 | 0.5236 | 0.5068 | 0.4911 | 0.4760 | 0.4618 | 0.4484 | 0.4356 | 0.4274 | 0.4235 |
RMLE | 2.0481 | 2.0481 | 2.0481 | 2.0481 | 2.0481 | 2.0481 | 2.0481 | 2.0481 | 2.0481 | 2.0481 | 2.0481 | 2.0481 |
LLE | 0.4235 | 0.4381 | 0.4531 | 0.4682 | 0.4836 | 0.4993 | 0.5153 | 0.5316 | 0.5481 | 0.5650 | 0.5803 | 0.5821 |
SRMLE | 0.4454 | 0.4454 | 0.4454 | 0.4454 | 0.4454 | 0.4454 | 0.4454 | 0.4454 | 0.4454 | 0.4454 | 0.4454 | 0.4454 |
k, d = 0.0 | k, d = 0.1 | k, d = 0.2 | k, d = 0.3 | k, d = 0.4 | k, d = 0.5 | k, d = 0.6 | k, d = 0.7 | k, d = 0.8 | k, d = 0.9 | k, d = 0.99 | k, d = 1.0 | |
---|---|---|---|---|---|---|---|---|---|---|---|---|
MLE | 1.1056 | 1.1056 | 1.1056 | 1.1056 | 1.1056 | 1.1056 | 1.1056 | 1.1056 | 1.1056 | 1.1056 | 1.1056 | 1.1056 |
LRE | 1.1056 | 1.0302 | 0.9629 | 0.9025 | 0.8481 | 0.7989 | 0.7542 | 0.7135 | 0.6763 | 0.6422 | 0.6139 | 0.6109 |
RMLE | 2.0444 | 2.0444 | 2.0444 | 2.0444 | 2.0444 | 2.0444 | 2.0444 | 2.0444 | 2.0444 | 2.0444 | 2.0444 | 2.0444 |
LLE | 0.6109 | 0.6534 | 0.6975 | 0.7432 | 0.7905 | 0.8393 | 0.8898 | 0.9418 | 0.9955 | 1.0507 | 1.1017 | 1.1075 |
SRMLE | 0.6967 | 0.6967 | 0.6967 | 0.6967 | 0.6967 | 0.6967 | 0.6967 | 0.6967 | 0.6967 | 0.6967 | 0.6967 | 0.6967 |
k, d = 0.0 | k, d = 0.1 | k, d = 0.2 | k, d = 0.3 | k, d = 0.4 | k, d = 0.5 | k, d = 0.6 | k, d = 0.7 | k, d = 0.8 | k, d = 0.9 | k, d = 0.99 | k, d = 1.0 | |
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MLE | 10.6280 | 10.6280 | 10.6280 | 10.6280 | 10.6280 | 10.6280 | 10.6280 | 10.6280 | 10.6280 | 10.6280 | 10.6280 | 10.6280 |
LRE | 10.6280 | 5.7256 | 3.6179 | 2.5069 | 1.8466 | 1.4212 | 1.1308 | 0.9235 | 0.7703 | 0.6538 | 0.5714 | 0.5632 |
RMLE | 2.0416 | 2.0416 | 2.0416 | 2.0416 | 2.0416 | 2.0416 | 2.0416 | 2.0416 | 2.0416 | 2.0416 | 2.0416 | 2.0416 |
LLE | 0.5632 | 0.9868 | 1.5399 | 2.2227 | 3.0350 | 3.9769 | 5.0483 | 6.2494 | 7.5800 | 9.0402 | 10.4652 | 10.6300 |
SRMLE | 1.8827 | 1.8827 | 1.8827 | 1.8827 | 1.8827 | 1.8827 | 1.8827 | 1.8827 | 1.8827 | 1.8827 | 1.8827 | 1.8827 |
Best Estimator | n = 20 | |||
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SRMLE | ||||
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n = 50 | ||||
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SRMLE | ||||
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n = 75 | ||||
LLE | ||||
SRMLE | ||||
LRE | ||||
n = 100 | ||||
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SRMLE | ||||
LRE |
n = 20 | n = 50 | ||
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SRMLE |
n = 75 | n = 100 | ||
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SRMLE |